Nuprl Lemma : eq-0-in-vs-quotient
∀[K:CRng]. ∀[vs:VectorSpace(K)]. ∀[P:Point(vs) ⟶ ℙ].
∀z:Point(vs). z = 0 ∈ Point(vs//z.P[z]) supposing ↓P[z] supposing vs-subspace(K;vs;z.P[z])
Proof
Definitions occuring in Statement :
vs-quotient: vs//z.P[z],
vs-subspace: vs-subspace(K;vs;x.P[x]),
vs-0: 0,
vector-space: VectorSpace(K),
vs-point: Point(vs),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
squash: ↓T,
function: x:A ⟶ B[x],
equal: s = t ∈ T,
crng: CRng
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
squash: ↓T,
vs-quotient: vs//z.P[z],
vs-0: 0,
vs-point: Point(vs),
mk-vs: mk-vs,
top: Top,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
bfalse: ff,
btrue: tt,
crng: CRng,
rng: Rng,
so_lambda: λ2x y.t[x; y],
so_lambda: λ2x.t[x],
so_apply: x[s],
so_apply: x[s1;s2],
implies: P ⇒ Q,
prop: ℙ,
eq-mod-subspace: x = y mod (z.P[z]),
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
rec_select_update_lemma,
istype-void,
quotient-member-eq,
vs-point_wf,
eq-mod-subspace_wf,
eq-mod-subspace-equiv,
vs-0_wf,
squash_wf,
vs-subspace_wf,
vector-space_wf,
crng_wf,
equal_wf,
vs-add-comm-nu,
vs-neg_wf,
iff_weakening_equal,
vs-add_wf,
vs-neg-zero,
vs-zero-add
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
sqequalHypSubstitution,
imageElimination,
sqequalRule,
extract_by_obid,
dependent_functionElimination,
thin,
isect_memberEquality_alt,
voidElimination,
hypothesis,
isectElimination,
setElimination,
rename,
hypothesisEquality,
lambdaEquality_alt,
because_Cache,
applyEquality,
inhabitedIsType,
independent_isectElimination,
independent_functionElimination,
universeIsType,
axiomEquality,
isectIsTypeImplies,
functionIsTypeImplies,
functionIsType,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
productElimination,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[K:CRng]. \mforall{}[vs:VectorSpace(K)]. \mforall{}[P:Point(vs) {}\mrightarrow{} \mBbbP{}].
\mforall{}z:Point(vs). z = 0 supposing \mdownarrow{}P[z] supposing vs-subspace(K;vs;z.P[z])
Date html generated:
2019_10_31-AM-06_27_53
Last ObjectModification:
2019_08_20-PM-05_57_57
Theory : linear!algebra
Home
Index